Previous to that, the special nature of these primes had been noted by Euclid, who showed that if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect. The first four primes of this form were known to him.

The fifth one, $M_{13}$, may have been known to Iamblichus in the $4$th century A.D.[1], but this is uncertain, as he does not explicitly demonstrate it. It was definitely known about by $1456$.

Pietro Cataldi is supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$. Recent researches[2], however, suggest that these may have already been discovered by $1460$. But as no evidence has been found from that date that they had been proven to be prime, it is possible that these were just lucky guesses.

Cataldi also claimed the primality of the Mersenne numbers $M_{23}, M_{29}, M_{31}$ and $M_{37}$.[3] In this he was correct only about $M_{31}$, so that also backs up the suggestion that he was only guessing.

$1903$: The factors of $M_{67}$ were found by Frank Cole who delivered a now famous lecture On The Factorization of Large Numbers in which he performed (without uttering a word) the arithmetic demonstrating what those factors were.

Thus Mersenne's assertion was finally investigated in full: he had been determined to be wrong by:

including $M_{67}$ and $M_{257}$ in his list of primes;

failing to include $M_{61}$, $M_{89}$ and $M_{107}$.

(Pervushin's discovery of the primality of $M_{61}$ caused some to suggest that Mersenne's claim of the primality of $M_{67}$ may have been a copying error for $M_{61}$.)

Nobody will ever know how Mersenne came to his conclusions, as it is impossible with the mathematical knowledge of the time for him to have worked it all out by hand. The fact that he made so few mistakes is incredible.